Growing Waves Due to Transverse Velocities By J. R. PIERCE and L. R. WALKER (Manuscript received March 30, 1955) This paper treats propagation of slaw ivavcs in two-dimensional neu- tralized electron flow in which all electrons have the same velocity in the direction of propagation but in which there are streams of two or more veloci- ties normal to the direction of propagation. In a finite beam in which electrons are reflected elastically at the boundaries and in which equal dc currents are carried by electrons with transverse velocities +Mi and — U\ , there is an antisymmctrical growing wave if co p 2 ~ (WHO* and a symmetrical growing wave if Here w p is plasma frequency for the total charge density and W is beam width. INTRODUCTION It is well-known that there can be growing waves in electron flow when the flow is composed of several streams of electrons having different velocities in the direction of propagation of the waves. 1 " 5 While Birdsall 6 considers the case of growing waves in electron flow consisting of streams which cross one another, the growing waves which he finds apparently occur when two streams have different components of velocity in the direction of propagation. This paper shows that there can be growing waves in electron flow consisting of two or more streams with the same component of velocity in the direction of wave propagation but with different components of velocity transverse to the direction of propagation. Such growing waves can exist when the electric field varies in strength across the flow. Such waves could result in the amplification of noise fluctuations in electron flow. They could also be used to amplify signals. 109 110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Actual electron flow as it occurs in practical tubes can exhibit trans- verse velocities. For instance, in Brillouin flow, 7 « 8 if we consider electron motion in a coordinate system rotating with the Larmor frequency we see that electrons with transverse velocities are free to cross the beam repeatedly, being reflected at the boundaries of the beam. The trans- verse velocities may be completely disorganized thermal velocities, or they may be larger and better-organized velocities due to aberrations at the edges of the cathode or at lenses or apertures. Two-dimensional Brillouin flow allows similar transverse motions. It would be difficult to treat the case of Brillouin or Brillouin-like flow with transverse velocities. Here, simpler cases with transverse velocities will be considered. The first case treated is that of infinite ion-neutra- lized two-dimensional flow with transverse velocities. The second case treated is that of two-dimensional flow in a beam of finite width in which the electrons are elastic-ally reflected at the boundaries of the beam. Growing waves are found in both cases, and the rate of growth may be large. In the case of the finite beam both an antisymmetric mode and a symmetric mode are possible. Here, it appears, the current density required for a growing wave in the symmetric mode is about % times as great as the current density required for a growing wave in the anti- symmetric mode. Hence, as the current is increased, the first growing waves to arise might be antisymmetric modes, which could couple to a symmetrical resonator or helix only through a lack of symmetry or through high-level effects. 1 . Infinite two-dimensional flow Consider a two-dimensional problem in which the potential varies sinusoidally in the y direction, as exp(-j/3z) in the z direction and as exp (jtd) with time. Let there be two electron streams, each of a negative charge p and each moving with the velocity u in the z direction, but with velocities «i and — Ui respectively in the y direction. Let us denote ac quantities pertaining to the first stream by subscripts 1 and ac quan- tities pertaining to the second stream by subscripts 2. The ac charge density will be denoted by p, the ac velocity in the y direction by y, and the ac velocity in the z direction by z. We will use linearized or small-signal equations of motion. 9 We will denote differentiation with respect to y by the operator D. The equation of continuity gives jcopi = -D(pi«i + poyi) + j/3(pit/o + Poii) (LI) jwp 2 = —D(-p 2 ih + Pom) + j/3(p2«o + P0Z2) (1.2) % GROWING WAVES DUE TO TRANSVERSE VELOCITIES 111 Let us define <h = j'(« - 0J/ O ) + u x D (1.3) (h = j(u - /3«o) - w,D (1.4) We can then rewrite (1.1) and (1.2) as <hp\ = Po(-Dyi + jfci) (1.5) d2pa = Poi-Dfa + jiara) (1.6) We will assume that we are dealing with slow waves and can use a po- tential V to describe the field. We can thus write the linearized equations of motion in the form diii = -j-PV (1.7) m d&. = -J-PV (1.8) m diyi = - DY (1.9) m chyi = - DV (1.10) in From (1.5) to (1.10) we obtain di'pi = --po(Z) 2 - /3 2 )7 (1.11) m r / 2 2 P2= -1 P0 {D 2 - p 2 )V (1.12) m Now, Poisson's equation is {If - tf)v = _^_±^. 2 (1.13) e From (1.11) to (1.13) we obtain (D 2 - /r')F = - V 2 ^ (JL + 1) (Z) 2 - ^ 2 )F (1.14) , ~ 2 J P ° (1.15) oj ;) — e Here co p is the plasma frequency for the charge of both beams. 112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Either (D 2 - /3 2 )7 = or else 1 = -up (d 2 + di) 2 di 2 d 2 2 We will consider this second case. We should note from (1.3) and (1.4) that d? = u{D 2 - (w - 0«o) 2 + 2jD(u - /3w )«i di = UiD* - (w - Quo? - 2jD(co - /3«o)mi d* + f /, 2 = 2[»! 2 Z) 2 - (to - |3wo) 2 ] dkW - [wi 2 D 2 + (- - /3«o) 2 ] 2 Thus, (1.17) becomes - <*,%& ~ (» - Z 3 ^) 2 ! [mi 2 Z> 2 + (co - 0-m o ) 2 ] 2 If the quantities involved vaiy sinusoidally with y as cos yy or sin yy, then (1.16) (1.17) (1.18) (1.19) (1.20) (1.21) (1.22) D 2 = -y- Our equation becomes 1 = J^L y-Ui 1 + 7Wi -~i 1 - yui J (1.23) (1.24) What happens if we have many transverse velocities? If we refer back to (1.14) we see that we will have an equation of the form i-E Vl^pn 2 f din + din din' din (1.25) Here o> pn 2 is a plasma frequency based on the density of electrons having transverse velocities ±u n . Equation (1.25) can be written y-u n - 1 + (« - /3i/o) 2 " 7 2 Mn 2 _ 1 - (CO - 0Mo) 2 ' ' 7 2 M n 2 (1-26) GROWING WAVES DUE TO TRANSVERSE VELOCITIES 113 (O>-/3U Fig. 1 Suppose we plot the left-hand and the right-hand sides of (1.26) versus (co — (iiio). The general appearance of the left-hand and right-hand sides of (1.2G) is indicated in Fig. 1 for the case of two velocities u n . There will always be two unattenuated waves at values of (w — /3w ) 2 > y 2 u e 2 where u e is the extreme value of ii„; these correspond to intersections 3 and 3' in Fig. 2. The other waves, two per value of u n , may be unat- tenuated or a pair of increasing and decreasing waves, depending on the values of the parameters. If > 1 there will be at least one pair of increasing and decreasing waves. It is not clear what will happen for a Maxwellian distribution of veloci- ties. However, we must remember that various aberrations might give a very different, strongly peaked velocity distribution. Let us consider the amount of gain in the case of one pair of transverse velocities, ±«i . The equation is now 2 2 7 "i >♦(* - 0woY' 7^1 / _ :>-(•- - 0ttoY" ' yui ) (1.27) Let « + .7^f U Wo (1.28) 114 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 0.9 0.7 0.5 0.3 0.2 0.1 Fig. This relation defines e. Equation (1.27) becomes yui 1 - € (1.29) «,* (1 + e 2 ) 2 In Fig. 2, e is plotted versus the parameter -fu*/wj. We see that as the parameter falls below unity, c increases, at first rapidly, and then more slowly, reaching a value of ±1 as the parameter goes to zero (as w p goes to infinity, for instance) . It will be shown in Section 2 of this paper that these results for infinite flow are in some degree an approximation to the results for flow in narrow beams. It is therefore of interest to see what results they yield if applied to a beam of finite width. If the beam has a length L, the voltage gain is e TC*i/*4£ (1.30) The gain G in db is (1.31) G = 8.7^edb wo GROWING WAVES DUE TO TRANSVERSE VELOCITIES 115 Let the width of the beam be W. We let y - w e- 32 > Thus, for n = 1, there is a half-cycle variation across the beam. From (1.31) and (1.32) G = 27.3 (-=\ru db (1.33) \Mo a / Now L/uq is the time it takes the electrons to go from one end of the beam to the other, while W/u\ is the time it takes the electrons to cross the beam. If the electrons cross the beam N times N ~lw wo Thus, G = 27.3 Nne db (1.35) While for a given value of e the gain is higher if we make the phase vary many times across the beam, i.e., if we make n large, we should note that to get any gain at all we must have 2 . 2 2 co p > 7 U\ 2 ^ /mrwiY (1-36) If we increase w p , which is proportional to current density, so that w p 2 passes through this value, the gain will rise sharply just after « p 2 passes through this value and will rise less rapidly thereafter. 2. A Two-Dimensional Beam of Finite Width. Let us assume a beam of finite width in the y-direction; the boundaries lying at y = ±?/o . It will be assumed also that electrons incident upon these boundaries are elastically reflected, so that electrons of the incident stream (1 or 2) are converted into those of the other stream (2 or 1). The condition of elastic reflection implies that fa = -2/2 (2.1) Zi = Z2 at y = ±?/o (2.2) and, in addition, that Pi = p 2 at y = ±?/o (2.3) since there is no change in the number of electrons at the boundary. 116 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The equations of motion and of continuity (1.7-1.12) may be satisfied by introducing a single quantity, \f/, such that V = dx diif (2.4) iy = —j — 8 d\ d% ty m (2.5) zi = — j — di dvj/ m (2.6) Vi = — di d^Dyf/ m (2.7) yi= — d\ drfty m (2.8) w = -- P0 (D 2 - 8 2 ) cfaV m (2.9) P2 = -1 UD 2 - 8 2 ) dxY (2.10) m Then, if we introduce the symbol, ft, for w — 8u yi + y2 = 2j - di. d 2 Dty (2.11) m h- 22 = 2j-did*uiDf (2-12) Pl - P2 = 2j- po(D 2 - 8 2 )uSlD+ (2.13) m It is clear that if Zty = 2>V = y = ±yo ( 2 - 14 ) the conditions for elastic reflection will be satisfied. The equation satis- fied by \f/ may now be found from Poisson's equation, (1-13), and is (D 2 - 8 2 ) d? d,V = — (& ~ /3 2 )(^i 2 + d?)+ me or (D 2 - 8 2 )[{u x 2 D 2 + ft 2 ) 2 + co p 2 ( Wl 2 D 2 - O 2 )] = (2.15) which is of the sixth degree in D. So far four boundary conditions have been imposed. The remaining necessary pair arise from matching the GROWING WAVES DUE TO TRANSVERSE VELOCITIES 117 internal fields to the external ones. For y > y V = Voe-^-e- " (2.16) and Similarly 5^ + (3V = at y = y oy ^ - 0V - at 7/ - -7/o (2.17) The most familiar procedure now would be to look for solutions of (2.15) of the form, e". This would give the sextic for c (c 2 - rtKWV + a 2 ) 2 + ^(Vc 2 - n 2 )] = o (2.18) with the roots c = ±/3, ±ci , ±c 2 , let us say. We could then express \f/ as a linear combination of these six solutions and adjust the coefficients to satisfy the six boundary equations. In this way a characteristic equa- tion for /3 would be obtained. From the symmetry of the problem this has the general form F(fi, c,) = F(/3, c 2 ), where a and c 2 are found from (2.18). The discussion of the problem in these terms is rather laborious and, if we are concerned mainly with examining qualitatively the onset of increasing waves, another approach serves better. From the symmetry of the equations and of the boundary conditions we see that there are solutions for ^ (and consequently for V and p) which are even in y and again some which are odd in y. Consider first the even solutions. We will assume that there is an even function, ^i(?/), periodic in y with period 2// , which coincides with \f/(y) in the open interval, — y < y < y n and that \pi(y) has a Fourier cosine series repre- sentation: 00 Hv) = E c n cos \ n y X„ = — n = 0, 1, 2, • • • (2.19) i 2/o i/> inside the interval satisfies (2.15), so we assume that rpi(y) obeys (D 2 - ?)[(*.?& + lf) J + o>;tu{D 2 - tf)fr + „ (220 > - E%- 2m + ly ) fn=— oo where 5 is the familiar 5-function. Since D\f/ and D'V are required to vanish at the ends of the interval and \f/, D 2 \f/ and Z)ty are even it follows that all 118 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 of these functions are continuous. We assume that i/-i = \f/, Ztyi = D\p, Dfyi = Dfy, -Dfyi = Z>V and Dfyi = ^V at tne en ^s of the intervals. From (2.20), u^Dfyi -> -V$ as jy -> y . Since £ 5(7/ - 2m + lft) = s?- + - L (-D B cos X„2/ (2.21) -oo *Z/<> I/O 1 we obtain from (2.20) 2ynP\ — —' W " Wp2) (2.22) . o y f-ir cos x - y ^ + V l (/3 2 + An 2 )[(a 2 - Wl 2 Xn 2 ) 2 - «,«(ff + Wl 2 Xn 2 )]/ Since |Z + pv = (D + Mui*D' + rf)V, using (2.4), the condition for matching to the external field, dV dy yields, using D$ - Z)V = and ifc'Dty = - W, the relation ( Wl 2 D 2 + rf)V» =3^ at y - yo . Applying this to (2.22), we then obtain, finally, Z/o _ 1 j§ /3 2 [n 2 - u p 2 ] (Q 2 - Ml V)' + 2E T - (0 2 + X„ 2 )[(fi 2 - Wl 2 X„ 2 ) 2 - co p 2 (fl 2 + M! 2 X„ 2 )] (2.23) For the odd solution we use a function, My), equal to \/f(y) in - j/ < y < ?/o and representable by a sine series. To ensure the vanishing of Zty and Dty at i/ = ±2/0 it is appropriate to use the functions, sin n„y, where Mn = ( n + H)t/i/o • The period is now 4y Q and we define My) in y a < y < 32/o by the relation ^2(2/) = tf(2j/o - 2/) and in -3y < y < -J/o by ^2(2/) = iV(— 2?/o — y)- Thus, we write CO ^2(2/) = S rfn Sin Mn2/ Mn = fa + MWft ^2(2/) will be supposed to satisfy GROWING WAVES DUE TO TRANSVERSE VELOCITIES 119 (D 2 - iflKnW + a 2 ) 2 + UrWl? - oft* ±» (2.24) = 2^ [5(2/ — 4m + I2/0) — 8(y — 4m — lf/o)] m= — oo The extended definition of fa (outside —y < y < ?y ) is such that we may again take xpi = \p,- • ■ ■ , Dfyi = DV at the ends of the interval. w/Dtyi is still equal to — }/£ at ?/ = // . Now +=0 H [8(.V - 4m + lyo) - £(y - 4m - lj/o)] (2.25) = — 2 (-1)" sin n„y so from (2.24) we may find ,,.,,, _ _V (-l) n sin M »/y r99(Vl {p l + Mn _ )l("" — ttlT*» r — Wp"("" + Ml Z Mn )J Matching to the external field as before gives (itiD'- + fl 2 )fy 2 = ss at ?/ =yo and applied to (2.26) we have oo /o 2 2 2 \ 2 -V± = V (n - Ml Mn ) , 2 2 _v 2/3 V (/3 2 + Mn 2 )[(fi 2 " WiVn 2 ) 2 - «,*<(* + MlVn*)] ' The equations (2.23) and (2.27) for the even and odd modes may be rewritten using the following reduced variables. .-ft 7T , COT/o Wo K = Z mil Wi 7T 2 Mi 2 (2.23) becomes /r 2 - S 2 £=[ 2 2 + n 2 (n 2 - A- 2 ) 2 - 5 2 (n 2 + /c 2 ) and (2.27) transforms to 2 J [(n + W - A- 2 ] 2 S s 2 + (n + y 2 y- [(n + ' 2 ) 2 - fc 2 ] 2 - 5 2 [(n + J 2 ) 2 + **] ( 2 . 2 9) = — 7T2 120 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 We shall assume in considering (2.28) and (2.29) that the beam is sufficiently wide for the transit of an electron from one side to the other to take a few RF cycles. The number of cycles is in fact, <ay /rui , and, hence, from the definition of z, we see that for values of k less than 2, perhaps, z is certainly positive. Let us consider (2.29) first since it proves to be the simpler case. If we transfer the term xz to the right hand side, it follows from the observa- tion that z is positive (for modest values of k), that it is necessaiy to make the sum negative. The sum may be studied qualitatively by sketch- ing in the k 2 — 5 2 plane the lines on which the individual terms go to infinity, given by <5 2 = [(n + y 2 ) 2 - iff (2.30) Fig. 3 GROWING WAVES DUE TO TRANSVERSE VELOCITIES 121 0.6 / 0.5 0.4 0."\ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 a/77 1.6 1. 2.0 2.2 2.4 Fig. 4 Fig. 3 shows a few such curves (n = 0, 1, 2). To the right of such curves the individual term in question is negative, except on the line, k 2 = (ft + \$) , where it attains the value of zero. Approaching the curves from the right the terms go to — oo . On the left of the curves the func- tion is positive and goes to + °° as the curve is approached from the ... /n = 2 /n=i /x\ = y z x < >< \ ^ 3 4 5 Fig. 5 122 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 left. Clearly in the regions marked + which lie to the left of every curve given by (2.30), the sum is positive and we cannot have roots. Let us examine the sum in the region to the right of the n = curve and to the left of all others. On the line, A; 2 = }>i, the sum is positive, since the first term is zero. On any other line, A: 2 = constant, the sum goes from + co at the n = 1 curve monotonically to — » at the n = curve, so that somewhere it must pass through 0. This enables us to draw the zero- sum contours qualitatively in this region and they are indicated in Fig. 3. We are now in a position to follow the variation in the sum as k varies at fixed 5 2 . It is readily seen that for 5 2 < 0.25, because —irz is negative in the region under consideration, there will be four real roots, two for positive, two for negative k. For 5 2 slightly greater than 0.25, the sum has Fig. 6A GROWING WAVES DUE TO TRANSVERSE VELOCITIES 123 a deep minimum for k = 0, so that there are still four real roots unless z is very large. For z fixed, as 8" increases, the depth of the minimum de- creases and there will finally occur a 8 2 for which the minimum is so shal- low that two of the real roots disappear. Call 2(0) the value of z for k = 0, write the sum as S(5 2 , fc 2 ) and suppose that S(5o 2 , 0) = —irz(0), then for small k we have S(S 2 , AT) = -«(0) + (6 2 - 8 2 ) §, + k 2 % = -«(0) -£ * do- dk~ Wo as as as aw afc 2 'as y a? — (6- - 5 -) + . 2 aP The roots become complex when 8 2 = 5 2 - (Ui/UoY as as as 2 ok- Since Wi/wo may be considered small (say 10 per cent) it is sufficient to look for the values of 5o . When k 2 = we have -7TZ = 22 (n + ht 2 2 + (n + ^)« (n + l AY ~ & * t(^4^ + z 2 + 5 2 V \(n + ^) 2 - 5 2 (n 4- H) 2 + * 2 (5 tan 7r5 + 2 tanh xz) z 2 + 5 Fig. 4 shows the solution of this equation for various 2(0) or coy /iruo • Clearly the threshold 5 is rather insensitive to variations in uyo/inio • Equation (2.28) may be examined by a similar method, but here some complications arise. Fig. 5 shows the infinity curves for n = 0, 1, 2, 3; the n = term being of the form k /k — 8 . The lowest critical region in 8 2 is the neighborhood of the point k = 8 = l /z, which is the intersec- tion of the n = and n = 1 lines. To obtain an idea of the behavior of 124 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 the left hand side (l.h.s.) of (2.28) in this area we first see how the point /,- 2 = 5 2 = }i can be approached so that the l.h.s. remains finite. If we put A; 2 = V3 + £ and a 1 = V3 + cc and expand the first two dominant terms of (2.28), then adjust c to keep the result finite as e — > we find - l 3g2 ~ 5 C " I 3z 2 + 1 c varies from - % to ! 4 as z goes from to 00 , changing sign at z~ = %. Every curve for which the l.h.s. is constant makes quadratic contact with the line 5" — V3 = c(k •- _ u ) a t k = 5 = V3. If we remember that the l.h.s. is positive for k s = 0, < 8' < 1 and for V = 1, < 5" < 1, n = o/^ SHADED AREAS jf NEGATIVE ff Jsk"' f 1 S S 1 n = i\ 1 3 Fig. 6B GROWING WAVES DUE TO TRANSVERSE VELOCITIES 125 since there are no negative terms in the sum for these ranges and again that the l.h.s. must change sign between the n = and n = 1 lines for any k 2 in the range < k 2 < 1 (since it varies from T <» to ± °°), this information may be combined with that about the immediate vicinity of S 2 = k 2 = % to enable us to draw a line on which the l.h.s. is zero. This is indicated in Figs. 6A and 6B for small z and large z respec- tively. It will be seen that the zero curve and, in fact, all curves on which the l.h.s. is equal to a negative constant are required to have a vertical tangent at some point. This point may be above or below k = V% (de- pending upon the sign of c or the size of z) but always at a 8 > V$. For 8 2 < )i there are no regions where roots can arise as we can readily see by considering how the l.h.s. varies with k 2 at fixed 8 2 . For a fixed 8 > \i we have, then, either for k 2 > \i or A; 2 < M, according to the size of z, a negative minimum which becomes indefinitely deep as 8 — » V§. Thus, since the negative terms on the right-hand side are not sensitive to small changes in 5 2 , we must expect to find, for a fixed value of the l.h.s., two real solutions of (2.28) for some values of 5 2 and no real solutions for some larger value of 5 2 , since the negative minimum of the l.h.s. may be made as shallow as we like by increasing 8 2 . By continuity then we expect to find pairs of complex roots in this region. Rather oddly these roots, which will exist certainly for 8 2 sufficiently close to V3 + 0, will disappear if S 2 is sufficiently increased. REFERENCES 1 L. S. Nergaard, Analysis of a Simple Model of a Two-Beam Growing-Wave Tube, RCA Review, 9, pp. 585-601, Dec, 1948. 2. J. R. Pierce and W. B. Hebenstreit, A New Type of High-Frequency Amplifier, B. S. T. J., 28, pp. 23-51, Jan., 1949. 3. A. V. Haeff, The Electron-Wave Tube — A Novel Method of Generation and Amplification of Microwave Energy, Proc. I.R.E., 37, pp. 4-10, Jan., 1949. 4. G. G. Macfarlane and H. G. Hay, Wave Propagation in a Slipping Stream of Electrons, Proc. Physical Society Sec. B, 63, pp. 409-427, June, 1950. 5. P. Guenard and H. Huber, Etude Expenmentale de L'Interaction par Ondes de Charge d'Espace au Sein d'Un Faisceau Electronique se Deplacant dans Des Champs Electrique et Magn^tique Crois^s, Annales de Radioelectricite, 7, pp. 252-278, Oct., 1952. 6. C. K. Birdsall, Double Stream Amplification Due to Interaction Between Two Oblique Electron Streams, Technical Report No. 24, Electronics Research Laboratory, Stanford University. 7. L. Brillouin, A Theorem of Larmor and Its Importance for Electrons in Mag- netic Fields, Phys. Rev., 67, pp. 260-266, 1945. 8. J. R. Pierce, Theory and Design of Electron Beams, 2nd Ed., Chapter 9, Van Nostrand, 1954. 9. J. R. Pierce, Traveling-Wave Tubes, Van Nostrand, 1950.